This manuscript describes applications of a reciprocity principle for magnetostatics in the design of permanent magnet systems. The contribution is timely and likely to be of interest to those developing small-scale magnetic resonance systems. The presentation of the work however is much more difficult to follow than it should be. The manuscript should be carefully edited to make it more reader-friendly. As submitted, I feel it is likely to be of use to those who are already expert in these applications, but with some rewriting could be of much broader utility.

Many of my concerns with the manuscript arise from the lack of clear statements of the problem being solved and the lack of clear definitions of assumptions, symbols, constraints etc. It feels as though the authors have a clear vision of what was done and how to do it, but as the manuscript is written, a reader who does not have this same vision struggles to follow.

Some more specific comments:

1) Terms like: anchor magnet, anchor volume, Btarget, minimum discernible thickness, etc, are used without definition. For a reader who is not intimately involved in this kind of design work, clear definitions when these terms are used would help.

2) In section 2.1.1 it would help to refer to Figure 1, and to state clearly what is being contemplated. The way this material is framed, there is a vector potential A, but no words are said about how A is created. It is then a confusing surprise to see B_rA which is apparently a remanant field associated with magnetic material that is associated with A. It might be much simpler to frame the problem more symmetrically from the start, perhaps labelling the domains with more descriptive names such as 'sample' and 'external,' or even just 1 and 2. Each of these regions could contain magnetic material and produce fields in the other domain? The 'anchor' could then be defined as a magnet in region 1 for design and/or fabrication purposes and 2 will contain the finished designed magnet. It is unclear to me why the vector potential is introduced at all here. It is never used, except as a label for one of the domains.

3) On line 81, the introduction of Btarget is unclear. What is B_target? Can the origin of the minus sign be explained simply?

4) Line 151: 'zero scaling law' What exactly is meant here?

5) Fig 4A: the coloured figure has many colours in it that aren't in the scale bar?

6) What are the different panels in Fig 4B? What are the x and y axes of these panels?

7) Please state clearly how the control actuation works? There is a hint in Figure 4 that they are moved up and down, though without any sense of scale it is unclear how far. How much force is required to move them and hold them in position?

8) It might help to clearly define the function and requirements of a profiling magnet.

9) Please explain the origin of Eq 8 more clearly and ensure that all quantities in it are clearly defined.

10) What is the dlength scale of T? I found: Due to the zero-scaling of the field profile, all dimensions were normalised to the outer diameter of the available design volume. Maybe a scale bar could go on Figure 4A?

11) I was unable to interpret Table 1. What is d?

12) line 216-217: "found to have a limiting performance" is very vague. Could you be more specific?

13) Fig 5: The upper x axis (eta) is non-monotonic? Evenly spaced tick-marks imply some quantity that is varying smoothly and uniformly. The eta axis is very misleading. It would be helpful to remind the reader what eta is.

14) Fig 5D inset. I don't understand what is being drawn at all.

15) As a non-expert in the work being done here, I found the presentation of the third application to be completely opaque. Only with some difficulty could I even follow the basic idea of what the goal was, and even so was left wondering at the end what the phrase:

"unbound magnetic field strength" (line 54) meant. Is the goal simply to maximize the field strength in that case?

16) Could the authors comment on how the structure in Figure 5 might be fabricated? The abstract does emphasize manufacturability.

17) I think it would help to mention the non-dimensionality of many of the variables much earlier in the paper, before it might cause confusion.

18) Figure 2C is unclear. How do the upper and lower x axes relate to each other? Why does the upper axis increase nonlinearly? Why is the region 1<d_M<1.5 the interesting region? Isn't d_M<1 the region of interest?

19) The caption to Figure 3 suggests that the structures under consideration are those of Fig 2, though Fig 3A appears to simulate an annular ring of magnetic powder rather than the discrete cylinders of Figure 2. 20) It would be helpful to give some sense of the scale of the calculations. Are these things that could be done in a few minutes on a desktop computer, or are they hours on a large cluster?

20) Many additional details could be provided. I do not think enough information is provided in the methods that these results could be reproduced.

I feel the manuscript needs to be edited thoroughly and many pieces re-written so that the paper communicates what is intended in a way that someone who doesn't already know can understand it.

This manuscript describes a method based on the reciprocity principle for analyzing and optimizing the design of permanent magnet systems for magnetic resonance (MR) experiments. The authors show that the reciprocity principle for magnetostatics can be used to simplify simulations of the performance of these systems, which are of great interest to people designing MR spectrometers based on permanent magnets. However, while the content of the manuscript will likely be of interest to readers, I feel the material is presented in a way that is somewhat opaque to those not already intimately familiar with the field. I feel the presentation of the content as it stands will present a barrier to understanding for non-experts, and I feel that with some work put into rewriting parts of the manuscript, this paper could be much more impactful. 

The paper begins with a description of the reciprocity principle for magnetostatics, and then moves through several examples where the reciprocity principle is used to analyze and/or optimize the performance of an array of permanent magnets. However, as far as I could follow, the manuscript lacked clear and adequate descriptions of the problems they were trying to solve. Part of the confusion stems from the fact that the paper is full of jargon unfamiliar to non-experts which is never defined. I recommend the authors rework the manuscript with an eye towards simplifying the presentation and clarifying the problems they are trying to address. Some particular points of clarification:

1. For non-experts, it would be helpful to see some dimensionful units. What are the units of B, for example? How physically large are these arrays? At the end of the manuscript it is explained that all quantities presented are dimensionless, however this should be made clear from the beginning. 
2. It is not clear to me how Equation 2 as written follows from Equation 1, if M, H, and B are defined as described in the preceding paragraph. Where do the self-interaction terms Ha*Ha and Hm*Hm come from? 
3. Figure 1:
1. The term “anchor magnet” is never clearly defined in the text.
4. Figure 2:
1. What is d_m? It seems either it is not clearly defined, or it is not used with a consistent definition.
2. Again, for non-experts, it would be helpful to see some dimensionful units here- what is the rough size of the B-field that can be produced by this microarray, for example?
3. What are the colors in Figure 2A) indicating?
4. As far as I can tell, B and C both show different aspects of the design assembled with and without the outer alignment structure, but the caption seems to indicate B and C deal with the two respective cases?
5. 2C is quite confusing to me. Are the d_m and alpha axes correlated? What exactly is being plotted here? 
5. Figure 3:
1. How is this self-alignment achieved?
2. Is there an intuitive reason why sigma_B shows the structure it does as a function of d_M? It is not clear to me what is being plotted here, what is the relationship between the magnet diameter and the anchor? 
6. Figure 4: 
1. A clearer presentation of the toroidal array would be helpful.
2. What is 4B showing? The caption is unclear, 4B is identified twice. Which plots show what?
3. How large are the T values presented? Again, some feeling for dimensions would be extremely helpful.
7. Figure 5:
1. What is 5D showing? What does the red represent? Why is the eta axis non-monotonic? 
8. There are several different notations used throughout the manuscript. Equations 1 and 2 use one vector notation, Equation 3 uses a second, different notation, Equations 4 and 5 use a third notation, and Equation 9 uses a forth notation.

The work described is very cool. I think one can say that Fig. 2 summarizes the main idea and procedure suggested. By using the principle of magnetostatic reciprocity, one can have magnets self-assemble into a configuration that will maximize the field in a specified region. Natually, this approach could be used for generating magnet designs that produce very homogeneous fields, or even a specific magnetic field distribution. The experimental and computational demonstrations will also be of high value in a range of magnet design applications.

I agree with the other comments, however, that the presentation is significantly non-optimal. Many terms are not explained well, and there are several inaccuracies, that make it verry difficult for the reader to follow. I therefore think that this manuscript will be significantly improved if all terms are carefully introduced and defined, and if the logical flow is improved.  

For example: 

(1) “This simplifies Eq. 1 to:” But Eq. 2 does not look like a simiplification, only if you know exactly what you are looking for. Perhaps rather this should be called a transformation?

(3) “Whereas this approach matches exactly the ideal coupling between two equal magnets,”

it is not clear what coupling between magnets means here

(4) “Assuming Btarget = −BrA in Eq. 1, one can see how maximum compliance with the target field is given by the minimum magnetostatic energy condition.” Taget field is not defined, and it is not clear where the minimum magnetostatic energy conditions comes from or what it refers to.

(5) Section heading “High-n applications” should probably be reworded to simply say something like “Arrangement of n magnets” (does it matter what kinds of magnets?)

(6) “which takes small results in neodymium magnets as μk = 1.03 and μ? = 1.12” what does small results mean?

(7) “Similarly, one can deduce a reciprocity between two regions in space and the magnets/fields contained therein.“  it is not clear whether this sentence indeed says what the authors wish to say.

(8) Convoluted sentence: “One can nonetheless estimate the relative inefficiency of a design in optimally placing magnetic energy in the anchor volume in which a sample will be placed. “

(9) “intensity-optimal magnet, which should be thought of as the ultimate goal of the MR magnet designer, as the field’s homogeneity can then be targeted through a wealth of solutions.” What is an intensity-optimal magnet and how do you balance this goal against the goal of homogeneity?

(10) “On the other hand, field inhomogeneity does not constitute a fundamental problem for NMR, as it only reduces the net measured signal, and not the local contributions. This effect comes naturally from local dephasing, which has successfully 320 been targeted with "shimming" RF pulses (Topgaard et al. (2004)), which can periodically or continually compensate for any “

(11) “local deviation in phase.” What type of phase is refered to?

Again, pretty cool work, but improvement in presentation will make it much more useful for a broad audience.